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Astrophysics · arXiv:astro-ph/0402403 v2 · submitted 17 Feb 2004 · 29 references

Electrodynamics of black hole magnetospheres

S. S. Komissarov

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The paper says

“Numerical simulations combined with simple analytical arguments are used to reach a number of important conclusions on the nature of the Blandford-Znajek mechanism.”

more of the abstract

We show that, just like in the Penrose mechanism and in the MHD models of Punsly and Coroniti, the key role in this mechanism is played by the black hole ergosphere. The poloidal currents are driven by the gravitationally induced electric field which cannot be screened within the ergosphere by any static distribution of the electric charge of locally created pair plasma. Contrary to what is expected in the Membrane paradigm, the energy and angular momentum are extracted not only along the magnetic field lines penetrating the event horizon but along all field lines penetrating the ergosphere. In dipolar magnetic configurations symmetric relative to the equatorial plane the force-free approximation breaks down within the ergosphere where a strong current sheet develops along the equatorial plane. This current sheet supplies energy and angular momentum at infinity to the surrounding force-free magnetosphere. The Blandford-Znajek monopole solution is found to be asymptotically stable and causal. The so-called horizon boundary condition of Znajek is shown to be a regularity condition at fast critical surface.

from the abstract

Pith says

Machine review · deepseek-v4.1-flash · 13 Nov 2026 · what was read was not recorded · not checked by a person

Reviewed in 2026 with later literature in view.

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Holds up, with conditions. The ergosphere theorem and the Znajek-as-regularity-condition reading are the real contributions; the 'current sheet supplies the energy' claim is an inference from local source signs, not a closed global budget. This line is wrong

3 major objections · 8 minor

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The paper says, at Abstract (s5); Sec. 6.1.2-6.1.3

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Pith says: Objection 1 of 3 · major

The abstract states that 'the energy and angular momentum are extracted ... along all field lines penetrating the ergosphere', but the analytical argument establishes only the angular-momentum/current half. Eq. (114) shows that inside the ergosphere, where alpha^2 - beta^2 < 0, the condition B^2 - D^2 > 0 requires H_phi != 0, i.e. the ergosphere forces a poloidal current.

more of objection 1

Energy flux, however, requires Omega != 0 (S_p = -H_phi Omega B_p, Eq. 83), and Sec. 6.1.2 explicitly constructs a consistent static solution with Omega = 0, H_phi != 0 and zero Poynting flux. The energy half therefore rests on the finite-inertia argument of Sec. 6.1.3, which is physically reasonable but is not a theorem in the sense of Sec. 6.1.1. The abstract and the conclusions should distinguish 'the ergosphere forces poloidal current' from 'the ergosphere forces energy extraction', or Sec. 6.1.3 should be strengthened into an argument that Omega cannot relax to zero in a realistic magnetosphere.

The paper says, at Abstract (s5); Eqs. (107)-(108); Secs. (50)-(51)

The review did not quote this passage.

Pith says: Objection 2 of 3 · major

The claim that the equatorial current sheet 'supplies energy and angular momentum at infinity to the surrounding force-free magnetosphere' is inferred from the signs of the local source terms W_t and W_phi in Eqs. (107)-(108) together with the conservation laws (50)-(51); it is not demonstrated as a global balance.

more of objection 2

A direct check would be to integrate -E.J and the angular-momentum source over the current-sheet volume and compare the result with the Poynting and angular-momentum fluxes through a large sphere, showing that the two agree in magnitude as well as sign. As written, only the sign of the sourcing is established, not its magnitude, and 'supplies' is an inference rather than a verified budget. This is load-bearing for the paper's central claim and should be added.

The paper says, at Sec. 5.4

The review did not quote this passage.

Pith says: Objection 3 of 3 · major

The paper reports that at higher grid resolution the ergospheric current sheet 'begins to show signs of unsteady behavior', and the author acknowledges that whether this is numerical or physical 'remains to be investigated'. Since the steady sourcing of the surrounding magnetosphere by the sheet is part of the central claim (abstract s4-s5), a resolution study is needed: report the …

more of objection 3

… sheet width and the integrated source magnitude at each refinement level, state what is converged and what is not, and quantify the residual time-dependence. Without this, the stability of the source, as distinct from the stability of the Blandford-Znajek monopole solution, is not established.

8 minor objections

The paper says, at Sec. 5.2

Pith says: minor objection 1 of 8

In the sentence beginning 'Matching of H_phi given by (104) with H_phi given by (104)', the second reference should be to Eq. (105), the Michel monopole result.

The paper says, at Sec. 6.1.1

Pith says: minor objection 2 of 8

The theorem is proved for remote sources that are non-rotating (Omega = 0), and the extension to rotating sources via Eq. (112) is qualitative ('highly unlikely to occur naturally'). It would be cleaner to state the theorem for the Omega = 0 case, which is the physically relevant one, and treat the rotating case as a separate remark with its own caveat.

show 6 more

The paper says, at Sec. 5.1

Pith says: minor objection 3 of 8

The Wald test is described only as a visual comparison ('these solutions are so close that one cannot see the difference'). A quantitative error measure (e.g. L1 norm of the flux-function difference versus resolution) would make the code-validation claim checkable.

The paper says, at Sec. 6.1.4

Pith says: minor objection 4 of 8

'Punsly and Coronity' should read 'Punsly and Coroniti'.

The paper says, at References

Pith says: minor objection 5 of 8

Jackson (1979) is listed twice, and the text cites Jackson (1975); please reconcile the citation and remove the duplicate entry.

The paper says, at Appendix A1, property (vi)

Pith says: minor objection 6 of 8

The notation switches between 'Theta' and 'theta' in the statement of the monotonicity property; please use one symbol consistently.

The paper says, at Fig. 4 caption

Pith says: minor objection 7 of 8

'There are 15 contour equally spaced' should read '15 contours'; the same caption should state which quantity is plotted in the middle panel (the magnetic flux function) explicitly.

The paper says, at Sec. 5.4 and Acknowledgments

Pith says: minor objection 8 of 8

Please check the typeset version for 'sings' (should be 'signs'), 'Indeen' (should be 'Indeed'), and 'anonimous' (should be 'anonymous').

What would settle it

Compute the steady-state energy and angular-momentum budget in the magnetospheric Wald simulation. Integrate -E.J over the ergospheric current-sheet volume and compare to the flux of S and L through a sphere at r ~ 10-30, separating field lines that cross the event horizon from those that only thread the ergosphere. If the ergosphere-only lines' flux is not balanced by the current-sheet source within ~20%, the 'supplies' claim (abstract:s5) fails and the energy must come from the horizon or numerical dissipation. Repeat at two grid resolutions (e.g. 400x500 and 800x1000) to check that the source integral converges; if it does not, the current-sheet sourcing is resolution-dependent.

from the skeptic's reading

How the review went

4 readings by deepseek-v4.1-flash · 13 Nov 2026

  1. Reader holds up, with conditions confidence high
  2. Skeptic partly agrees 2 min 51 s · 34,196 tokens read, 10,150 written
  3. Editor wrote the verdict above 18 s · 35,192 tokens read, 998 written
  4. Referee major revision: 3 major objections, 8 minor 2 min 16 s · 34,502 tokens read, 7,269 written

Not run on this paper: the simulated authors' defence.

Pith did not check this review's objections against the paper.

The review in full

The review in brief, machine-written

Worth your time, but read it for two things and be skeptical about a third. The two things: the 3+1 formulation in Sec 2, which is genuinely more general than Thorne-Macdonald (it holds in any stationary slicing, including Kerr-Schild, and the constitutive equations 29-30 are cleaner), and the reinterpretation of Znajek's horizon condition as a regularity condition at the fast critical surface (Sec 6.2, eq 118). That second point is the one that actually settles the Punsly-Coroniti causality objection, and it is correct: in force-free electrodynamics the fast surface sits on the horizon, so Znajek's condition is a regularity requirement, not a boundary condition. The theorem in Sec 6.1.1 is also clean — eq (111) is a two-line argument and it holds. Within the ergosphere alpha2 - beta2 < 0, so no static charge distribution can screen the induced field. That is a real result and it is what makes the ergosphere, not the horizon, the driver. The numerical work backs the qualitative picture: the magnetospheric Wald problem does develop an equatorial current sheet inside the ergosphere, and field lines threading the ergosphere do rotate with the hole (Fig 4, Fig 5). Now the third thing, and this is where I would push back on the abstract. The angular-momentum half of the claim is a theorem; the energy half is not. Sec 6.1.2 itself constructs a consistent static solution with Omega = 0, H_phi != 0, and zero Poynting flux — the ergosphere forces poloidal current, not energy extraction. Energy extraction needs Omega != 0, which comes from the finite-inertia argument of Sec 6.1.3, physically reasonable but not proved. Eqs (107, 108) give positive local source terms; 'supplies' in abstract:s5 is an inference from those signs plus conservation, not an integrated budget over the sheet compared to the flux through a large sphere. That integration is the missing step. On the resistivity worry: less load-bearing than it looks. sigma_perp sets the sheet width, not the sign of Wt or W_phi, so the qualitative sourcing survives the hand-tuned parameters. The author is honest about the unphysical resistivity (Sec C3) and about the sheet going unsteady at higher resolution (Sec 5.4); no convergence study, and he says so. For a 2004 paper that is acceptable, but a referee should ask for the global energy balance. Send it out.

The strongest objection

The central claim (abstract:s3, s5) has two halves. The angular-momentum half is established analytically: Sec 6.1.2, eq (114), shows that inside the ergosphere alpha^2 - beta^2 < 0, so B^2 - D^2 > 0 requires H_phi != 0; the ergosphere forces poloidal current. The energy half is weaker. Energy flux requires Omega != 0 (S_p = -H_phi Omega B_p, eq 83), and the paper's own Sec 6.1.2 constructs a consistent static solution with Omega = 0, H_phi != 0, and zero Poynting flux. So the ergosphere forces current, not energy extraction; energy extraction follows only from the finite-inertia argument of Sec 6.1.3, which is physically reasonable but not a theorem. The simulation (Fig 4 left, Fig 5) shows Omega != 0 and H_phi != 0 along ergosphere-penetrating lines, but the paper never integrates the source -E.J over the current sheet and compares it to the flux through a large sphere. Eqs (107,108) give local positive source terms; 'supplies' (s5) is an inference from local signs plus the conservation law, not a demonstrated global balance. The authors also report the current sheet becomes unsteady at higher resolution (Sec 5.4) and present no convergence study, so the stability of the source is open. The reader's concern about the resistivity parameters is related but less load-bearing: sigma_perp sets the sheet width, not the sign of W_t or W_phi, so the qualitative sourcing claim is robust to that choice.

The paper is a careful and influential study. The 3+1 reformulation (Sec 2) is genuinely more general than Thorne-Macdonald and works in Kerr-Schild coordinates. The theorem in Sec 6.1.1 is a clean geometric argument: within the ergosphere alpha^2 - beta^2 < 0, so no static charge distribution can screen the gravitationally induced electric field. The reinterpretation of Znajek's condition as a regularity condition at the fast critical surface (Sec 6.2, eq 118) is a real conceptual clarification. The numerical results (Fig 4, 5) support the qualitative picture: the magnetospheric Wald problem develops an equatorial current sheet inside the ergosphere, and all ergosphere-penetrating field lines rotate with the hole. My concern is not that the central picture is wrong; it is that the specific claim that the current sheet supplies the fluxes (abstract:s5) is supported by local source-term signs and nonvanishing fluxes, but not by a global budget. The paper's own Sec 6.1.2 shows that the ergosphere forces H_phi != 0 but not Omega != 0, so the energy-extraction half of the claim depends on the finite-inertia argument of Sec 6.1.3 and on the simulation. The authors acknowledge the current sheet becomes unsteady at higher resolution (Sec 5.4) and present no convergence study. These are acknowledged limitations, but they bear directly on the sourcing claim, so they are worth stating as the load-bearing soft spot. The reader's verdict of CONDITIONAL is appropriate; my concern sharpens the reason rather than changing the verdict. The reader's weakest_assumption points at the resistivity parameters, which affect the sheet width but not the sign of the source; the more load-bearing gap is the missing global budget and the Omega != 0 dependence.

The referee's report

If the central claim holds, it reframes the Blandford-Znajek mechanism away from the Membrane Paradigm's horizon-as-unipolar-inductor picture and aligns it with the Penrose mechanism and with Punsly and Coroniti's emphasis on the ergosphere, while rebutting their causality objection. The 3+1 formulation of Sec. 2 is a genuine and independently useful contribution: it is more general than the Thorne-Macdonald system and is written in a form suited to Godunov-type codes in horizon-penetrating coordinates. The reinterpretation of Znajek's condition as a regularity condition at the fast critical surface (Sec. 6.2, Eqs. 117-118) is a clean result that resolves a long-standing confusion. The code tests against Wald's vacuum solution (Sec. 5.1) and the Blandford-Znajek perturbative monopole (Sec. 5.2), together with the Alfven-wave and current-sheet Riemann tests of Appendix C3, are good practice and give the numerical results credibility. The paper also states its limitations explicitly (Sec. 5.4 on current-sheet unsteadiness, Sec. C3 on the resistivity model, Sec. 7 on particle inertia), which is to its credit.

Pith's reading of the paper (a machine's summary, not the authors')

Komissarov asks where a spinning black hole's rotational energy is actually picked up, and answers that it is the ergosphere, the region where no observer can stand still, rather than the event horizon. He first rewrites the 3+1 equations of black hole electrodynamics so that they look like ordinary Maxwell equations in matter and stay regular on the horizon in Kerr-Schild coordinates, then runs time-dependent simulations with those equations. The simulations reproduce Wald's vacuum solution, recover the Blandford-Znajek monopole solution from two very different starting states, and show that a dipolar magnetosphere settles into a state with a strong current sheet along the equatorial plane inside the ergosphere. The argument that carries the paper is short: inside the ergosphere the gravitationally induced electric field is always stronger than the magnetic field, so no static charge distribution can screen it, and poloidal currents must flow. If that is right, the familiar picture of field lines being torqued by a conducting horizon is at best incomplete.

What carries the argument. The load-bearing object is the gravitationally induced electric field inside the ergosphere, written through the shift vector as $D = -frac{1}{alpha},beta times B$. It is the black-hole analogue of a pulsar's rotationally induced field, but it cannot be cancelled by any static arrangement of charge because the ergosphere makes $alpha^2-beta^2$ negative. Supporting machinery: a rewritten 3+1 system in which gravity enters only through the constitutive relations $E = alpha D + beta times B$ and $H = alpha B - beta times D$; a light-surface analysis showing that the inner light surface always lies between the ergosphere and the horizon and coincides with the inner Alfvén surface in the force-free limit, which is what makes the ergosphere causally connected to the outgoing wind; and a radiative resistivity built from inverse Compton scattering of background photons, which lets the code resolve current sheets by holding $B^2 - D^2$ near zero instead of letting it go negative.

Core claim. The central claim is that the Blandford-Znajek mechanism is driven by the ergosphere, not by the horizon. Komissarov argues that no steady-state axisymmetric vacuum field supported by distant, slowly rotating sources can satisfy both $Dcdot B = 0$ and $B^2 - D^2 > 0$ along field lines that penetrate the ergosphere: there $alpha^2-beta^2 < 0$, so $B^2 - D^2 = frac{alpha^2-beta^2}{alpha^2},B^2 < 0$ however strong the field is. The unscreened electric field drives cross-field currents, then charge separation, then poloidal currents that generate $H_varphi$. Consequently, in the magnetospheric Wald problem with a dipolar field, all field lines entering the ergosphere are forced to rotate with the hole whether or not they cross the horizon, and both energy and angular momentum flow outward along them. Where the configuration is symmetric about the equatorial plane, $H_varphi$ must vanish there, $B^2 - D^2 > 0$ cannot hold, and the force-free approximation breaks down; the equatorial current sheet that forms inside the ergosphere then acts as the localized source of the energy and angular momentum carried by the surrounding force-free magnetosphere. The paper also reports that the Blandford-Znajek monopole solution is asymptotically stable in time-dependent simulations, and that Znajek's horizon condition is not a boundary condition but a regularity condition at the fast critical surface of the ingoing wind.

The premise it rests on. The claim that the equatorial current sheet supplies the magnetosphere's energy and angular momentum rests on a resistivity model whose coefficients are chosen by hand and which the author states is much larger than the physical one, together with the assumptions of marginal screening $B^2 = D^2$ and rotation slower than the local zero-angular-momentum observer.

If the paper is right (Pith's inferences, not the authors')

Field lines that thread the ergosphere but miss the horizon still power outflows, so jet-power estimates that count only horizon flux will undershoot.

The Membrane Paradigm's stretched-horizon circuit becomes an incomplete analogy; the ergosphere, not the horizon, is where the 'battery' sits.

The Blandford-Znajek monopole solution is a genuine asymptotic attractor of the time-dependent equations, which defuses the Punsly-Coroniti causality objection to it.

Znajek's horizon condition is reclassified as a regularity condition at the fast critical point, so steady-state wind solutions are fixed by regularity rather than by horizon physics.

Dissipative current sheets inside the ergosphere are a generic feature of symmetric dipolar magnetospheres, so force-free codes must carry some resistivity to model them.

If the ergosphere is what matters, a non-rotating hole should extract nothing electromagnetically even when threaded by a strong externally supported field, which gives a clean null test separating ergospheric from horizon-based pictures.

The current sheet's width and location should track the pair-production rate and the soft-photon energy density, so the sheet could be a diagnostic of the ambient radiation field rather than of spin alone.

The result points away from the battery analogy and toward a wave-launching picture: the ergosphere couples to the outgoing wind through fast and Alfvén waves, which time-dependent simulations could test by tracking where the poloidal current system first establishes itself.

If the force-free approximation necessarily fails in the equatorial plane inside the ergosphere, then ideal-MHD jet-launching models that assume force-free conditions all the way to the horizon are missing a structural ingredient near the equator.

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The paper

Abstract

Numerical simulations combined with simple analytical arguments are used to reach a number of important conclusions on the nature of the Blandford-Znajek mechanism. We show that, just like in the Penrose mechanism and in the MHD models of Punsly and Coroniti, the key role in this mechanism is played by the black hole ergosphere. The poloidal currents are driven by the gravitationally induced electric field which cannot be screened within the ergosphere by any static distribution of the electric charge of locally created pair plasma. Contrary to what is expected in the Membrane paradigm, the energy and angular momentum are extracted not only along the magnetic field lines penetrating the event horizon but along all field lines penetrating the ergosphere. In dipolar magnetic configurations symmetric relative to the equatorial plane the force-free approximation breaks down within the ergosphere where a strong current sheet develops along the equatorial plane. This current sheet supplies energy and angular momentum at infinity to the surrounding force-free magnetosphere. The Blandford-Znajek monopole solution is found to be asymptotically stable and causal. The so-called horizon boundary condition of Znajek is shown to be a regularity condition at fast critical surface.

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. No sentence in this record was written by a model.

abstract compiled · 2026-09-23

The authors' main claim, as the classifier picked it out 0.51

“Contrary to what is expected in the Membrane paradigm, the energy and angular momentum are extracted not only along the magnetic field lines penetrating the event horizon but along all field lines penetrating the ergosphere.”

How the main claim is stated: asserted 1.00

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.29 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.94 · mentions code or data availability 0.02

The abstract, sentence by sentence

This table labels the abstract only (arXiv listing abstract v2). The number is the probability the classifier gave the label.

SentenceRoleThe authors' own workStates the main conclusionStrength
Numerical simulations combined with simple analytical arguments are used to reach a number of important conclusions on the nature of the Blandford-Znajek mechanism. method 0.93 0.89 0.06 asserted 0.96
We show that, just like in the Penrose mechanism and in the MHD models of Punsly and Coroniti, the key role in this mechanism is played by the black hole ergosphere. result 0.96 0.85 0.31 asserted 1.00
The poloidal currents are driven by the gravitationally induced electric field which cannot be screened within the ergosphere by any static distribution of the electric charge of locally created pair plasma. result 0.92 0.76 0.16 asserted 1.00
Contrary to what is expected in the Membrane paradigm, the energy and angular momentum are extracted not only along the magnetic field lines penetrating the event horizon but along all field lines penetrating the ergosphere. result 0.99 0.83 0.32 asserted 1.00
In dipolar magnetic configurations symmetric relative to the equatorial plane the force-free approximation breaks down within the ergosphere where a strong current sheet develops along the equatorial plane. result 1.00 0.83 0.10 asserted 1.00
This current sheet supplies energy and angular momentum at infinity to the surrounding force-free magnetosphere. result 0.80 0.80 0.13 asserted 1.00
The Blandford-Znajek monopole solution is found to be asymptotically stable and causal. result 1.00 0.91 0.14 asserted 1.00
The so-called horizon boundary condition of Znajek is shown to be a regularity condition at fast critical surface. result 0.86 0.89 0.10 asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.95 · evidence from simulation or computation 1.00

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Figures (6)
Figure 1 of the paper
Figure 1. Schwarzschild black hole in uniform magnetic field. The black circle shows the position of the event horizon and the continuous lines show the magnetic flux surfaces. νb = eBˇ mW c (91) is the electron gyration frequency, νc = (W2 − 1)σT ub mW v (92) is the effective frequency of the inverse Compton collisions; here m is the particle rest mass. The last term in eq.(88) does not depend on the sign of the electric charge and describes the usual drift motion effected by the collisions. Equations (87,88) give us the generalized Ohm law of the same form as eq.(85) with σk = ne2 m
Figure 2 of the paper
Figure 2. Monopole field solution for a = 0.1 and B0 = 1 at time t = 50. Panel a) shows the angular velocity of magnetic field lines at r=3. The perturbative solution of Blandford and Znajek gives Ω = 0.5Ωh, where Ωh is the angular velocity of the black hole. Panel b) shows Hφ of the numerical solution (the crosses) and of the perturbative solution (the continuous line) at r = 3. Panel c) shows Eˇ ·Bˇ along θ = 1 and θ = π − 1. The small unscreened component Eˇk of electric field drives the conductivity current towards the black hole in the upper hemisphere and away from it in the lower hemisp
Figure 3 of the paper
Figure 3. Escape of the split-monopole magnetic field from a Schwarzschild black hole. Left panel: Magnetic flux surfaces of the split￾monopole solution which was used as an initial solution in these numerical simulations. Right panel: Magnetic flux surfaces of the numerical solution at t = 5. so far and for this reason it has been playing a key role in the development of the black hole electrodynamics. One im￾portant property of this solution is that all magnetic field lines penetrate the black hole horizon. Macdonald (1984) at￾tempted to construct numerical steady-state solutions for a more…
Figure 4 of the paper
Figure 4. Magnetospheric Wald problem. Left panel: The angular velocity of magnetic field lines. There are 15 contours equally spaced between 0 and 0.67. The angular velocity first gradually increased towards the axis but then reaches a maximum and goes slightly down. The thick lines show the ergosphere (the outer line) and the inner light surface (the inner line). Middle panel: The magnetic flux surfaces. Right panel: The distribution of (B2 − D2 )/max(B2 , D2 ). There are 15 contour equally spaced between -0.12 and 1.0. This quantity monotonically decreases towards the current sheet in the…
Figure 5 of the paper
Figure 5. The distribution of the angular velocity of magnetic lines (left panel) and Hφ (right panel) in the magnetospheric Wald problem at r = 1.8. Similar calculations lead to the expression for the angular momentum source, Wφ, in the equatorial plane Wφ = −(J ×B)·m = ασ⊥B 2 . (108) Thus, provided the magnetosphere rotates slower then ZAMO, the resistive dissipation in the ergospheric current sheet can provide positive sources of electromagnetic energy and angular momentum. Since, the inner light surface en￾closes the current sheet (see the left panel of figure 4) the condition Ω < Ωz is
Figure 6 of the paper
Figure 6. Left panel: The initial currents driven by the gravitationally induced electric field within the ergosphere as it is being filled with pair plasma during the thought experiment described in Sec.6.1.1. Right panel: The electric current system of the steady-state Wald’s magnetosphere. that simultaneously satisfy both D·B = 0 and B 2 − D 2 > 0 along the magnetic field lines penetrating the ergosphere of a rotating black hole. To prove this statement let us assume that electric cur￾rents of these sources can be arranged in such a way that D·B = 0 everywhere in the vacuum zone surroundin

Figures are the authors', reproduced from the paper.

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Cited by 15 Pith papers

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  5. Force-free electrodynamics near rotation axis of a Kerr black hole gr-qc · Aug 2019
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  7. Destinies of Plasmoids Formed from the Magnetic Reconnection Occurring within the Ergo Region of a Rotating Black Hole gr-qc · Oct 2026
  8. Model dependent analytic spin torsion corrections to Blandford Znajek energy extraction in Einstein Cartan gravity physics.gen-ph · Aug 2026
  9. Unveiling the electrodynamic nature of spacetime collisions gr-qc · Apr 2025
  10. Mining Gravitational-wave Catalogs To Understand Binary Stellar Evolution: A New Hierarchical Bayesian Framework astro-ph.HE · Jun 2018
  11. A Measurement of the Electromagnetic Luminosity of a Kerr Black Hole astro-ph · Apr 2004
  12. Distinct Near-Horizon Trend of Synchrotron Polarization in Kerr Spacetime gr-qc · Jun 2026
  13. Comisso-Asenjo Mechanism in Rotating $\mathcal{N}=2,U(1)^2$ Gauged Supergravity Black Holes: Extended Comparison With Kerr Black Hole gr-qc · Jun 2026
  14. GW170817, General Relativistic Magnetohydrodynamic Simulations, and the Neutron Star Maximum Mass astro-ph.HE · Nov 2017
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How later papers used it

78 later papers cite it

Evidence: substantial. 78 of the citing papers share no author with it, and 18 of those carry a confident label from Pith's citation classifier.

Pith read 18 of the 159 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 5 of 14 papers with this label
Compares with it 1 of 4 papers with this label
How the literature uses it

36 arXiv papers cite this in 75 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 46 sentences without a confident label

Uses its method or tool 6 papers
  • Using the vacuum constitutive equations derived by Komissarov [ref] , [equation] and setting $β^i = 0$, we obtain [equation] Thus the vacuum behaves as an isotropic effective medium with scalar permittivity and permeability: [equation] The distinction from the usual stationary optical-medium analogy is important: here $ϵ$ and $μ$ depend on time through $m(r,t)$.

    Electromagnetic Emission from a Black Hole Evaporating in External Magnetic Field · Electromagnetic Emission from an Evaporating Black Hole: A Macroscopic Medium Perspective · 0.91

  • If we now interpret $𝒰^μν_α̂$ as a field strength tensor in its own right, we are led to introduce auxiliary electric and magnetic fields, [equation] which resemble electromagnetic fields in a medium, and are related to $B^α̂μ$ and $E^α̂μ$ via constitutive relations [ref] , [equation] We can recast Eqs. [ref], [ref] exactly in a more familiar vector calculus form [ref] , also following [ref] , [equation] where $√(γ) = √(-g)/α$, and $α$ and $β$ are the gravitational lapse and shift, respectively [ref] .

    Unveiling the electrodynamic nature of spacetime collisions · Introduction · 0.81

  • This amounts to imposing the force-free conditions [ref] [equation] We impose these conditions in the exterior of the neutron stars using the relaxation approach of [ref] , i.e. [equation] where $ν^-1= 1.25×10^-6 Ω^-1$ is a short relaxation time, $χ$ is the Heaviside function, and $Ω$ is the orbital angular speed, see Tab. [ref].

    Electromagnetic precursor flares from the late inspiral of neutron star binaries · Force-free electrodynamics · 1.00

Uses its data or values 1 paper
Compares with it 4 papers
Cites it as support 1 paper
Cites it as background 10 papers

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